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06 Oct 2020, 05:45
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D
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53% (03:14) correct
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ABCD is a square and E is the midpoint of AD. A circle is drawn through the points E, B, and C as shown above. What the ratio of the perimeter of the square to the perimeter of the circle?
A. \(\frac{\pi}{4}\)
B. \(\frac{5\pi}{16}\)
C. 1:1
D. \(\frac{16}{5\pi}\)
E. \(\frac{4}{\pi}\)
Are You Up For the Challenge: 700 Level Questions
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Untitled.png [ 42.65 KiB | Viewed 4314 times ]
06 Oct 2020, 10:16
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It's a pure ratio question, so we can just make up a length. So we can say the side of the square is length 2. Say also the radius of the circle is r.
If we draw a vertical line through the middle of the square from point E downwards, cutting the square in half, that line will be of length 2. The top part of that line will be a radius, so will be length r. The bottom part of that line will thus be of length 2-r.
Now if we connect the center of the circle to point C, thus drawing a radius, we'll get a small right triangle at the bottom of the square. Its hypotenuse is the radius, so is length r. As we saw above, the vertical leg is of length 2-r, and the horizontal leg is half of the side of the square, so is length 1. Thus by Pythagoras,
r^2 = 1^2 + (2 - r)^2
r^2 = 1 + 4 - 4r + r^2
4r = 5
r = 5/4
So the circumference of the circle is 2πr = 5π/2, and the perimeter of the square is (4)(2) = 8, and the ratio the question asks for is equal to 8 / (5π/2) = 16/5π.
If we draw a vertical line through the middle of the square from point E downwards, cutting the square in half, that line will be of length 2. The top part of that line will be a radius, so will be length r. The bottom part of that line will thus be of length 2-r.
Now if we connect the center of the circle to point C, thus drawing a radius, we'll get a small right triangle at the bottom of the square. Its hypotenuse is the radius, so is length r. As we saw above, the vertical leg is of length 2-r, and the horizontal leg is half of the side of the square, so is length 1. Thus by Pythagoras,
r^2 = 1^2 + (2 - r)^2
r^2 = 1 + 4 - 4r + r^2
4r = 5
r = 5/4
So the circumference of the circle is 2πr = 5π/2, and the perimeter of the square is (4)(2) = 8, and the ratio the question asks for is equal to 8 / (5π/2) = 16/5π.
06 Oct 2020, 17:53
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Bunuel
Set the side length of the square as \(s\) and the radius of the circle as \(r\). Draw the center of the circle and complete as much information as we can (see attachment). The goal is to find a relation between s and r, we can build a triangle with the given information and use the Pythagorean theorem to get:
\((\frac{s}{2})^2 + (s - r)^2 = r^2\)
\(\frac{1}{4}s^2 + s^2 - 2sr + r^2 = r^2\)
\(\frac{5}{4}s^2 - 2sr = 0\)
\(\frac{5}{4}s - 2r = 0\) (s = 0 is not an interesting solution so we can just cancel s)
\(s = 2r*\frac{4}{5} = \frac{8r}{5}\)
\(\frac{s}{r} = \frac{8}{5}\)
Finally we want the ratio of the perimeters, which is:
\(\frac{4s}{2\pi*r} = \frac{4}{2\pi} *\frac{8}{5} = \frac{16}{5\pi}\).
Ans: D
Attachments
Circle+Square.png [ 55.51 KiB | Viewed 3761 times ]
